I know that commutative rings and division rings have the invariant basis number property. I'm curious what else are there.
Do Noetherian rings have the invariant basis number property?
If not, what about Artinian rings or semisimple rings? And how do I prove this?
Let $R\neq\{0\}$ be Noetherian and let $m > n$ be natural numbers. Consider the homomorphism $f ~:~ R^m \to R^n$ given by the projection onto the first $n$ coordinates.
The assumption $R^m\cong R^n$ leads to a contradiction, as $f$ has non-zero kernel but it is also a surjective endomorphism of a Noetherian module, hence bijective (this is easily proved using the ascending chain condition).